= Cartier-divisor description of the Picard group
{c}
{title2=$\operatorname{Pic}(X)\cong\Gamma(X,\mathcal K_X^*/\mathcal O_X^*)/k(X)^*$}
On an <irreducible variety>, a <Cartier divisor> is a section of $\mathcal K_X^*/\mathcal O_X^*$: its local rational equations differ by regular units. Their ratios define an <invertible sheaf>, and a global rational equation gives the trivial class. Conversely a rational trivialization of an <invertible sheaf> gives local equations. The rational-function sheaf is <flasque>, so its <long exact sequence in sheaf cohomology> also identifies the quotient with $H^1(X,\mathcal O_X^*)$.
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